Bosonic code 

Also known as Continuous-variable (CV) quantum code, Oscillator code, Quantum modulation scheme.
Root code for the Bosonic Kingdom

Description

Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode).

States of a single oscillator correspond to \(L^2\)-normalizable functions on \(\mathbb{R}\) that have finite energy, finite variance, and finite values of all other moments (where the energy operator is defined to be the harmonic oscillator Hamiltonian); such functions form Schwartz space, a subspace of Hilbert space [1]. Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.

States can be represented by a series via a basis expansion, such as that in the countable basis of Fock states \(|n\rangle\) with \(n\geq 0\). Alternatively, states can be represented as functions over the reals by expanding in a continuous "basis" (more technically, set of tempered distributions in the space dual to Schwartz space), such as the position "basis" \(|y\rangle\) with \(y\in\mathbb{R}\) or the momentum "basis" \(|p\rangle\) with \(p\in\mathbb{R}\). A third option is to use coherent states \(|\alpha\rangle\) with \(\alpha\in\mathbb{C}\), which are eigenstates of the annihilation operator, which correspond to classical electromagnetic signals, and which resolve the identity [2–5]. States can further be represented as functions over the joint position-momentum phase space in the Wigner function formalism [6,7].

An important subset of states is formed by the Gaussian states, which are in one-to-one correspondence with a (displacement) vector and covariance matrix [8–12]. Pure Gaussian states can be obtained from the vacuum Fock state \(|n=0\rangle\) via a Gaussian unitary transformation (defined below). Any coherent state can be obtained from the vacuum Fock state, itself a coherent state, by a displacement.

Protection

Displacement error basis

An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes.

Displacement operators: For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states \(|y\rangle\) for \(y\in\mathbb{R}\) as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \tag*{(1)}\end{align} where \(q\in\mathbb{R}\). The former is also called a translation, while the latter is called a modulation in signal processing. For multiple modes, error set elements are tensor products of elements of the single-oscillator error set, characterized by the vector of coefficients \(\xi\in\mathbb{R}^{2n}\).

The displacement error set is a unitary basis for bounded operators on the \(n\)-mode Hilbert space that is Dirac-orthonormal under the Hilbert-Schmidt inner product. Expanding a bounded operator in terms of displacements is called the Fourier-Weyl transform (a.k.a. Fourier-Weyl relation) [14][13; Eq. (4.11)]. For the expansion of Gaussian unitary operations in terms of displacements, see [15; Eq. (7.62)].

There are two definitions of code distance associated with displacements. The definition inherited from qubit codes is the minimum weight of a displacement operator (i.e., number of nonzero entries in \(\xi\)) that implements a nontrivial logical operation in the code. The second definition is the minimum Euclidean distance (i.e., \(\ell^2\)-norm of \(\xi\)) such that the corresponding displacement implements a nontrivial logical operation in the code.

Loss and gain operators

An error set relevant to Fock-state bosonic codes is the set of loss operators associated with the AD channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set.

Number-phase operators

An related error set is the set of powers of the Susskind–Glogower phase operator \(\frac{1}{\sqrt{a a^\dagger}} a\) and its adjoint [16–18] along with Fock-space rotations generated by the occupation number operator \(a^\dagger a\). These can also be obtained from qudit Pauli matrices through a limiting procedure [18] and allow one to expand trace-class operators despite not forming an orthonormal set [1]. These operators are correspong to the number-phase interpretation, a polar-like decomposition of a single mode, complementing the cartesian-like decomposition in terms of position and momentum displacements.

Rate

The quantum capacity of the AD channel [19] and the dephasing noise channel [20] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [21,22]. Exact two-way assisted capacities have been obtained for the AD channels and quantum limited amplifiers in what is known as the PLOB bound [23]. These are examples of Gaussian channels, i.e., channels that map Gaussian states to Gaussian states [24–30]. Non-Gaussian channel capacities can be bounded for single [31] and multiple [32; Lemma 14] modes.

Gates

Displacement operations form a group called the Heisenberg-Weyl group, the oscillator analogue to the Pauli group. Analogues of (non-Pauli) Clifford-group transformations are the Gaussian unitary transformations (a.k.a. symplectic, Bogoliubov-Valatin, or linear canonical transformations) [10,33,34], which are unitaries generated by quadratic polynomials in positions and momenta. The Gaussian unitary transformation group permutes displacement operators amongst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{R})\).Computing using Gaussian states and Gaussian unitaries only can be efficiently simulated on a classical computer [35–37]. This remains true even if superpositions of Gaussian states are considered [38,39], but is no longer the case when the number of modes scales exponentially [40]. A cubic phase gate is required to make a universal gate set on the oscillator [8,41]; other gates are possible, but cubic or higher versions of squeezing are not well defined [42]. See Ref. [43] for bosonic computational complexity classes.Controllability of bosonic states has been proven when the normalizable state space is restricted to Schwartz space [44].Measurements can be performed by homodyne, heterodyne, and generalized homodyne measurements [45].The number-phase interpretation allows for the mapping of rotor Clifford gates into the oscillator, some of which become non-unitary (e.g., conditional occupation number addition) [46].ZX calculus has been extended to bosonic codes for both Gaussian operators [47] and Fock-state based operators [48]. An earlier graphical calculus exists for Gaussian pure states [49].Circuits can be decomposed into a series of primitives such as quantum lattice gates [50].

Notes

For an introduction to continuous-variable quantum systems, see reviews [1,51–55] and books [13,56,57].See video tutorial by V. V. Albert.

Parent

  • Group-based quantum code — Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.

Children

Cousins

  • \(t\)-design — The notion of quantum state designs has been extended to states of a bosonic mode [58].
  • Bosonic c-q code — Bosonic c-q codes are bosonic codes designed to transmit classical information.
  • EA bosonic code — EA bosonic codes utilize additional ancillary modes in a pre-shared entangled state, but reduce to ordinary bosonic codes when said modes are interpreted as noiseless physical modes.
  • Fermion code — Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.

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“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oscillators
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@incollection{eczoo_oscillators, title={Bosonic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/oscillators} }
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